NP-hardness conjecture for $\mathrm{PCSP}(\mathbf{1in3},\mathbf{\check{C}^+})$

Let $\mathbf{1in3}$ and $\mathbf{\check{C}^+}$ be the finite relational templates used in the paper, with a homomorphism $\mathbf{1in3}\to\mathbf{\check{C}^+}$. The associated promise constraint satisfaction problem is $\mathrm{PCSP}(\mathbf{1in3},\mathbf{\check{C}^+})$. $\mathbf{\check{C}^+}$ hardness conjecture. $\mathrm{PCSP}(\mathbf{1in3},\mathbf{\check{C}^+})$ is NP-hard. The paper presents this as an unresolved four-element case and provides evidence for, but does not establish, the claimed hardness.